Controlled mobility in stochastic and dynamic wireless networks
The MIT Faculty has made this article openly available.
Please share
how this access benefits you. Your story matters.
Citation
Çelik, Güner D., and Eytan H. Modiano. “Controlled Mobility in
Stochastic and Dynamic Wireless Networks.” Queueing Systems
72.3-4 (2012): 251–277.
As Published
http://dx.doi.org/10.1007/s11134-012-9313-y
Publisher
Springer-Verlag
Version
Author's final manuscript
Citable link
http://hdl.handle.net/1721.1/81444
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Detailed Terms
http://creativecommons.org/licenses/by-nc-sa/3.0/
(will be inserted by the editor)
Controlled Mobility in Stochastic and Dynamic Wireless Networks
G ¨uner D. C¸ elik · Eytan H. Modiano
Invited Paper
Abstract We consider the use of controlled mobility in wireless networks where messages arriving
randomly in time and space are collected by mobile receivers (collectors). The collectors are responsi-ble for receiving these messages via wireless communication by dynamically adjusting their position in the network. Our goal is to utilize a combination of wireless transmission and controlled mobility to improve the throughput and delay performance in such networks. In the first part of the paper we consider a system with a single collector. We show that the necessary and sufficient stability condition for such a system is given by ρ < 1 where ρ is the average system load. We derive lower bounds
for the average message waiting time in the system and develop policies that are stable for all loads
ρ < 1 and have asymptotically optimal delay scaling. We show that the combination of mobility and
wireless transmission results in a delay scaling ofΘ(1−ρ1 ) with the system load ρ in contrast to the Θ( 1
(1−ρ)2) delay scaling in the corresponding system where the collector visits each message loca-tion. In the second part of the paper we consider the system with multiple collectors. In the case where simultaneous transmissions to different collectors do not interfere with each other, we show that the stability condition is given byρ < 1, where ρ is the system load on multiple collectors. We develop
lower bounds on delay and generalize policies established for the single collector case to multiple collectors case. We show that the delay scaling ofΘ( 1
1−ρ) extends to the case of multiple collectors, in contrast to theΘ((1−ρ)1 2) delay scaling in the corresponding multi-collector system without wire-less transmission. We also consider the case where simultaneous transmissions to different collectors interfere with each other. We characterize the stability region of the system in terms of interference constraints. We show that a frame-based version of the well-known Max-Weight policy is throughput-optimal asymptotically in the frame length and derive an upper bound on average delay under this policy.
Keywords Spatial Queueing Models· Controlled Mobility in Wireless Networks · Dynamic Vehicle
Routing· DTRP · Delay Tolerant Networks G ¨uner D. C¸ elik
Massachusetts Institute Of Technology
77 Massachusetts Avenue, 31-011, Cambridge, MA, 02139 Tel.: +1 (617) 230-1558
E-mail: gcelik@mit.edu Eytan H. Modiano
Massachusetts Institute Of Technology
77 Massachusetts Avenue, 33-412, Cambridge, MA, 02139 Tel.: +1 (617) 452-3414
Collector
< r∗
Message
λ
Fig. 1: The system model for the case of a single collector. The collector adjusts its position in order to receive randomly arriving messages via wireless communication. The circles with radiusr∗represent the communication range and the dashed lines represent the collector’s path.
1 Introduction
There has been a significant amount of interest in performance analysis of mobility assisted wire-less networks in the last decade (e.g., [21], [23], [33], [32], [41], [43] [44], [49], [50], [52]). Typically, throughput and delay performance of networks were analyzed where nodes moving according to a ran-dom mobility model were utilized for relaying data (e.g., [20, 21, 23, 32, 39]). More recently, networks deploying nodes with controlled mobility have been considered focusing primarily on route design and ignoring the communication aspect of the problem (e.g., [13], [21], [33], [44], [49], [26], [50], [53]). In this paper we explore the use of controlled mobility and wireless transmission in order to improve the throughput and delay performance of wireless networks. We consider a dynamic vehicle routing problem where a vehicle (collector) uses a combination of physical movement and wireless reception to receive randomly arriving data messages.
Our model consists of collectors that are responsible for gathering messages that arrive randomly in time at uniformly distributed geographical locations. The messages are transmitted when a col-lector is within their communication distance and depart the system upon successful transmission. Collectors adjusts their positions in order to successfully receive these messages in the least amount of time as shown in Fig. 1 for the case of one collector. This setup is particularly applicable to net-works deployed in a large area so that mobile elements are necessary to provide connectivity between spatially separated entities in the network [13], [26], [33], [52]. For instance, this model is applicable to a densely deployed sensor network where mobile base stations collect data from a large number of sensors densely deployed inside the network, [27], [33], [44], [52], [53]. Another application is utilizing Unmanned Aerial Vehicles (UAVs) as data harvesting devices or as communication relays on a battlefield environment [18], [41], [26], [53]. This model also applies to networks in which data rate is relatively low so that data transmission time is comparable to the collector’s travel time, for instance in underwater sensor networks [1], [45].
Vehicle Routing Problems (VRPs) have been extensively studied in the past (e.g., [2], [6], [8], [9], [10], [17], [18], [34], [49], [50], [51]). The common example of a VRP is the Euclidean Traveling
Salesman Problem (TSP) in which a single server is to visit each member of a fixed set of locations on the plane such that the total travel cost is minimized. Several extensions of TSP have been considered in the past such as stochastic demand arrivals and the use of multiple servers [2], [8], [9], [18]. In particular, in the TSP with neighborhoods (TSPN) problem the vehicle is to visit a neighborhood of each demand location [6], [17], [34], which can model a mobile collector receiving messages from a communication distance. A more detailed review of the literature in this field can be found in [9], [34] and [51].
Of particular relevance to us among the VRPs is the Dynamic Traveling Repairman Problem (DTRP) due to Bertsimas and van Ryzin [8], [9], [10]. DTRP is a stochastic and dynamic VRP in which a vehicle is to serve demands that arrive randomly in time and space. Fundamental lower bounds on delay were established and several vehicle routing policies were analyzed for DTRP for a single server in [8], for multiple servers in [9], and for general demand and interarrival time distributions in [10]. Altman and Levy [2] considered a similar problem termed queuing in space and and proposed stabilizing algorithms. Later, [49], [50] generalized the DTRP model to analyze Dynamic Pickup and Delivery Problem (DPDP) where fundamental bounds on delay were established. We apply the DTRP model to wireless networks where the demands are data messages to be transmitted to a collector which is capable of wireless communication1. In our system the problem has considerably different
characteristics since in this case the collector does not have to visit message locations but rather can receive the messages from a distance using wireless communication. The objective in our system is to effectively utilize this combination of wireless transmission and controlled mobility in order to minimize the time average message waiting time.
In a closely related problem where multiple mobile nodes with controlled mobility and communi-cation capability relay the messages of static nodes, [43] derived a lower bound on node travel times. Message sources and destinations are modeled as static nodes in [43] and these nodes have saturated arrivals hence queuing aspects were not considered. In an independent work, [27] considered utiliz-ing mobile wireless servers as data relays on periodic routes and applied various delay relations from Polling models to this setup. A mobile server harvesting data from spatial queues in a wireless net-work was considered in [41] where the stability region of the system was characterized using a fluid model approximation. In [15] we analyzed a one-collector model similar to the current paper but for which the arriving messages were transmitted to the collector using a random access scheme, creating interference among neighboring transmissions. In this paper, the message transmissions are sched-uled, i.e., there is only one transmission in the system at a given time, and the collector decides on the message to be transmitted next. The two systems have considerably different characteristics as will be explained in the following sections.
Another related body of literature lies in the area of utilizing mobile elements that can control their mobility to collect sensor data in Delay Tolerant Networks (DTN) (e.g., [13, 33, 44, 45, 52, 53]). Route selection (e.g., [33], [44], [53]), scheduling or dynamic mobility control (e.g., [13], [45], [52]) algorithms were proposed to maximize network lifetime, to provide connectivity or to minimize delay. More detailed surveys of the related work in the area of utilizing mobility in DTN and Sensor Net-works can be found in [45] and [53]. These Net-works focus primarily on mobility and usually consider particular policies for the mobile element. To the best of our knowledge, this is the first attempt to de-velop fundamental bounds on delay in a system where a collector is to gather data messages randomly arriving in time and space using wireless communication and controlled mobility.
In the first part of the paper we consider a system with a single collector and extend the results of [8] for the DTRP problem to the communication setting. In particular, we show that ρ < 1 is
the necessary and sufficient condition for the stability of the system whereρ is the system load. We
derive lower bounds on delay and develop algorithms that are asymptotically within a constant factor of the lower bounds. We show that the combination of mobility and wireless transmission results in a delay scaling ofΘ(1/(1 − ρ)) in contrast to the Θ(1/(1 − ρ)2) delay scaling in the system where
1 In previous works such as [2], [8], [9], the collector needs to be at the message location in order to be able to serve it, therefore, we will refer to the DTRP model as the system without wireless transmission.
the collector visits each message location analyzed in [2], [8]. In the second part of the paper we consider the system with multiple collectors under the assumption that simultaneous transmissions to different collectors do not interfere with each other. We show that the necessary and sufficient stability condition is still given byρ < 1, where ρ is the load on multiple collectors. We develop fundamental
lower bounds on delay in the system and generalize the single-collector policies analyzed in the first part to the multiple collectors case. Finally we consider a multiple-collector system under interference constraints on simultaneous transmissions to different collectors. We formulate a scheduling problem and characterize the stability region of the system in terms of interference constraints on simultaneous transmissions. We show that a frame-based version of the seminal Max-Weight scheduling policy can stabilize the system whenever it is stabilizable and we derive an upper bound on average delay under this policy.
This paper is organized as follows. In Section 2 we consider the single collector case. We present the model in Section 2.1, characterize the necessary and sufficient stability condition in Section 2.2, derive the delay lower bound in Section 2.3, and analyze single-collector policies in Section 2.4. In Section 3 and the subsections therein we extend the results for a single collector to systems with multiple collectors whose transmissions do not interfere with each other. In Section 4 we consider the system with interference constraints on simultaneous transmissions to collectors. We first present the model and characterize the stability region, and then analyze the frame based Max-Weight policy in Section 4.1 and propose an upper bound on the delay performance of this policy in Section 4.2.
2 Single Collector
In this section we consider the case of a single collector and develop fundamental insights into the problem. We extend the stability and the delay results in [2] and [8], established for the system where the collector visits each message location, to systems with wireless transmission capability. We show that the combination of mobility and wireless transmission results in a delay scaling ofΘ( 1
1−ρ) with the system loadρ in contrast to the Θ((1−ρ)1 2) delay scaling in the corresponding system without wireless transmission in [2] and [8].
2.1 Model
Consider a square regionR of area A and messages arriving into R according to a Poisson process (in time) of intensityλ. Upon arrival the messages are distributed independently and uniformly in R and they are to be gathered by a collector via wireless reception. An arriving message is transmitted to the collector when the collector comes within the reception distance of the message location and grants access for the message’s transmission. Therefore, there is no interference power from the neighboring nodes during message receptions.
We assume a Disk Model (or communication range model) [16], [24] for determining successful message receptions. Letr∗be the reception distance of the collector. Under the disk model, a trans-mission can be received only if it is within a disk of radiusr∗around the collector. Note that the Disk Model is similar to the Signal to Noise Ratio (SNR) packet reception model [16], [23], [24], termed the SNR Model, under which a transmission is successfully decoded at the collector if it’s received SNR is above a thresholdβ. To see this, if PT is the constant transmit power level of a transmis-sion at distancer away from the collector, due to distance-attenuation, the received power satisfies PR= PTr−α[16], [23], [24], whereα is the power loss exponent. Therefore, under the SNR Model, a transmission at distancer to the collector is successful if r ≤ r∗ = (P. T/(PNβ))1/α, showing the equivalence to the Disk Model. Under the Disk Model, if the location of the next message to be re-ceived is withinr∗, the collector stops and attempts to receive the message. Otherwise, the collector travels towards the message location until it is within a distancer∗away from the message. Under the
disk model, transmissions are assumed to be at a constant rate taking a fixed amount of time denoted bys.
The collector travels from the current message reception point to the next message reception point at a constant speedv. We assume that at a given time the collector knows the locations and the arrival
times of the messages that arrived before this time. The knowledge of the service locations is a standard assumption in vehicle routing literature [2], [6], [8], [17], [18], [29], [34], [49].
LetN (t) denote the total number of messages in the system at time t.
Definition 1 (Stability [5], [35], [37]) The system is stable under a given control policyπ if lim sup
t→∞ E[N (t)] <∞,
namely, the long term expected number of messages in the system is finite. Letρ = λs denote the load
arriving into the system per unit time. For stable systems,ρ denotes the fraction of time the collector
spends receiving messages.
Definition 2 (Stability Region [37], [38], [46]) The stability region Λ is the set of all loadsρ such
that there exists a control algorithm that stabilizes the system.
A policy is said to be throughput-optimal if it stabilizes the system for all loads strictly inside Λ. We defineTi as the time between the arrival of messagei and its successful reception. Ti has three components:Wd,i, the waiting time due to collector’s travel distance from the time messagei arrives until it gets served, Ws,i, the waiting time due to the reception times of messages received from the time messagei arrives until it gets served, and s, reception time of the message. The total
waiting time of messagei is denoted by Wi = Wd,i+ Ws,i, henceWi = Ti− s. We let dibe the collector travel distance from the collector’s reception location for the message served prior to message
i to collector’s reception location for message i. The time average per-message travel distance of the
collector, denoted byd, is defined by an expectation in the steady state given by d = limi→∞E[di]. The time average delaysT , W , WdandWsare defined similarly to haveT = Wd+ Ws+ s whenever the limits exist.T∗is defined to be the optimal system time which is given by the policy that minimizes
T .
2.2 Stability
In this section we show that ρ < 1 is a necessary and sufficient condition for the stability of the
system. Note that this condition is also necessary and sufficient for stability of the corresponding system without wireless transmission, as shown in [2], as well as for a G/G/1 queue [28]. Here we prove this result using simpler techniques than [2]. The analysis in this section will be essential for generalizing the stability condition and some delay results to the case of multiple collectors.
2.2.1 Necessary Condition for Stability
We lower bound the number of messages in the system by that in the equivalent system in which travel times are zero (i.e.,v = ∞). This technique was used in [2] to establish a necessary stability condition for the corresponding system without wireless transmission. Here we give a simpler proof of this fact in Appendix A for completeness.
Theorem 1 A necessary condition for stability isρ < 1. Furthermore, we have
W ≥ λs
2
The proof in Appendix A first establishes that the steady state time average delay in the system under any policyπ is at least as big as the delay of any work-conserving2policy in the equivalent system
in which travel times are zero (i.e.,v = ∞). This is based on an induction argument that the total number of messages in the system is always greater than that in the infinite velocity system. This is because the service time per message is greater than that in the infinite speed system. Since the latter system behaves as an M/D/1 queue (a queue with Poisson arrivals, constant service times and 1 server), its average waiting time is given by the Pollaczek-Khinchin (P-K) formula for M/G/1 queues [7, p. 189], given in (1). A direct consequence of this lemma is that a necessary condition for stability in the infinite speed system is also necessary for our system. The necessary and sufficient condition for stability in an M/G/1 queue is given byρ < 1 (see e.g., [7] or [22]).
2.2.2 Sufficient Condition for Stability
Here we prove thatρ < 1 is a sufficient condition for stability of the system under a policy based on
Euclidean TSP with neighborhoods (TSPN). TSPN is a generalization of TSP in which the server is to visit a neighborhood of each demand location via the shortest path [6], [17], [34]. In our case the neighborhoods are disks of radiusr∗around each message location. TSPN is an NP-Hard problem such as TSP. Recently, [34] proved that a Polynomial Time Approximation Scheme (PTAS) exists for TSPN among fat regions in the plane. A region is said to be fat if it contains a disk whose size is within a constant factor of the diameter of the region, e.g., a disk, and a PTAS belongs to a family of
(1 + )-approximation algorithms parameterized by > 0.
Under the TSPN policy, the collector performs a cyclic service of the messages present in the system starting and ending the cycle at the center of the network region. Let timetkbe the time that the collector returns to the center for thekth time, where t0= 0. Assume the system is initially empty. at timet0. The TSPN Policy is described in detail in Algorithm 1.
Algorithm 1 TSPN Policy
1: Initially att = t0, the collector waits at the center ofR until the first message arrival, moves to serve this message and returns to the center.
2: If the system is empty at timetk, k = 1, 2, ..., the collector repeats the above process.
3: If there are messages waiting for service at timetk, k = 1, 2, ..., the collector computes the TSPN tour (e.g., using the PTAS in [34]) through all the messages that are present in the system at time
tk, receives these messages in that tour and returns to the center.
Let the total number of messages waiting for service at timeti,N (ti), be the system state at time
ti. Note thatN (ti) is an irreducible Markov chain on countable state space N. We show the stability of the TSPN policy through the ergodicity of this Markov chain.
Theorem 2 The system is stable under the TSPN policy for all loadsρ < 1.
Proof Given the system stateN (ti) at time ti, we apply the algorithm in [34] to find a TSPN tour of lengthLithrough theN (ti) neighborhoods that is at most (1 + ) away from the optimal TSPN tour lengthL∗
i. Note thatL∗i can be upper bounded by a constantL for all N (ti). This is because the collector does not have to move for messages within its communication range and a finite number of such disks of radiusr∗ can cover the network region for anyr∗ > 0. The collector then can serve the messages in each disk from its center incurring a tour of constant lengthL (an example of such
a tour is shown in Fig. 2). We will use the Foster-Lyapunov criterion to show that the Markov chain described by the statesN (ti) is positive recurrent [5]. We use V (Ni) = sN (ti), the total load served
duringithcycle, as the Lyapunov function. Note thatV (0) = 0, S
k = {x : V (x) ≤ K} is a bounded set for all finite K and V (.) is a non-decreasing function. Since the arrival process is Poisson, the
expected number of arrivals during a cycle can be upper-bounded as follows:
E[N (ti+1)|N(ti)] ≤ λ(L/v + sN(ti)). (2) Hence we obtain the following drift expression for the load during a cycle.
E[sN (ti+1) − sN(ti)|N(ti)] ≤ ρL/v − (1 − ρ)sN(ti). (3) Sinceρ < 1, there exist a δ > 0 such that ρ + δ < 1:
E[sN (ti+1) − sN(ti)|N(ti)] ≤ ρL/v − δsNi
≤ −δs +ρLv .1{N(ti)∈S}, (4) where 1{N∈S}is equal to1 if N ∈ S and zero otherwise and S = {N ∈ N : N ≤ K} is a bounded set withK = dvδsρL + 1e. Hence the drift is negative as long as N(ti) is outside a bounded set. Therefore, by the standard Foster-Lyapunov criterion [3], [5], the Markov chain (N (ti)) is positive recurrent and it has a unique stationary distribution [5]. Furthermore, using (3) we can bound the steady state averageN (ti) as [35]
lim sup
ti→∞
E[N (ti)] ≤ λL
v(1 − ρ).
Moreover, given somet ∈ [ti, ti+1], we have
lim sup
t→∞ E[N (t)]≤ lim supti→∞
E[N (ti) + N (ti+1)]
≤ 2 λL
v(1 − ρ) < ∞. (5)
This establishes the stability of the TSPN policy for any loadρ < 1. ut
This establishes thatρ < 1 is a necessary and sufficient condition for stability. The travel time
of the collector does not affect the stability region of the system. In other words, we have the same stability region as a G/G/1 queue. The intuition behind this result is that as the system load is increased, under stable policies, the fraction of time spent on travel goes to zero. This observation is also true for the corresponding system without wireless transmission sinceρ < 1 is also necessary and sufficient
for stability of such systems [2].
The communication capability does not enlarge the stability region, however, it fundamentally affects the delay scaling in the system. The delay scaling of the TSPN policy with loadρ is O(1−ρ1 )
as shown in (5), the same delay scaling as in a G/G/1 queue. This is a fundamental improvement in delay due to the wireless transmission capability as the delay scaling for the corresponding system without wireless transmission isΘ( 1
(1−ρ)2) [8]. This delay scaling can be easily obtained as follows. The system without wireless transmission corresponds to havingr∗ = 0 in our system. In this case, one utilizes a (1 + ) PTAS for the optimal TSP tour through the message locations instead of the
TSPN tour. An upper bound on the TSP tour for anyN (ti) points arbitrarily distributed in a square of areaA is given byp2AN(ti) + 1.75
√
A [29]. Similar arguments as in the proof of Theorem 2 leads
to the drift condition
E[sN (ti+1) − sN(ti)|N(ti)] ≤ ρ(κ1pAN(ti) + κ2) − (1 − ρ)sN(ti), (6) for some constantsκ1andκ2, where the drift is again negative as long asN (ti) is outside a bounded setS. The difference in this case is that the travel time per cycle scales with the number of messages N (ti) aspN(ti) and using (6) we can show that the delay scaling with the load ρ is O((1−ρ)1 2).
2.3 Lower Bound On Delay
For wireless networks with a small area and/or very good channel quality such that r∗ ≥ pA/2, the collector does not need to move as every message will be in its reception range if it just stays at the center of the network region. In that case the system can be modeled as an M/D/1 queue with service times and the associated queuing delay is given by the P-K formula for M/G/1 queues, i.e., W = λs2/(2(1 − ρ)). However, when r∗ < pA/2, the collector has to move in order to receive some of the messages. In this case the reception times is still a constant, however, the travel time per
message is now a random variable which is not independent over messages (for example, observing small travel times for the previous messages implies a dense network, and hence the future travel times per message are also expected to be small). Next we provide a lower bound similar to a lower bound in [8] with the added complexity of communication capability in our system.
Theorem 3 The optimal steady state time average delayT∗is lower bounded by
T∗≥E[(||U|| − r∗)
+]
v(1 − ρ) + λs2
2(1 − ρ)+ s. (7)
where(||U|| − r∗)+representsmax(0, ||U|| − r∗), U is a uniformly distributed random variable over
the network regionR, and ρ = λs is the system load.
Note that theE[(||U|| − r∗)+] term can be further lower bounded by E[||U||] − r∗, whereE[||U||] =
0.383√A [8].
Proof As outlined in Section 2.1, the delay of messagei, Ti has three components:Ti = Wd,i+
Ws,i+ si. Taking expectations and the limit asi → ∞ yields
T = Wd+ Ws+ s. (8)
A lower bound on Wd is found as follows: Note that Wd,i.v is the average distance the collector moves during the waiting time of messagei. This distance is at least as large as the average distance
between the location of messagei and the collector’s location at the time of message i’s arrival less the
reception distancer∗. The location of an arrival is determined according to the uniform distribution over the network region, while the collector’s location distribution is in general unknown as it depends on the collector’s policy. We can lower boundWdby characterizing the expected distance between a uniform arrival and the best a priori location in the network that minimizes the expected distance to a uniform arrival. Namely we are after the locationν that minimizes E[||U −ν||] where U is a uniformly distributed random variable. The locationν that solves this optimization is called the median of the
region and in our case the median is the center of the square shaped network region. Because the travel distance is nonnegative, we obtain the following bound onWd:
Wd≥ E[(||U|| − r ∗)+]
v . (9)
LetN be the average number of messages received in a waiting time and let R be the average residual
reception (service) time. Due to the PASTA property of Poisson arrivals (Poisson Arrivals See Time Averages) (see for example [7, p. 171]) a given arrival in steady state observes the time average steady state occupancy distribution. Therefore, the average residual time observed by an arrival is alsoR and
is given byλs2/2 [7, p. 188] and we have
Ws= sN + R. (10)
Since in a stable system in steady state the average number of messages received in a waiting time is equal to the average number of arrivals in a waiting time (a variation of Little’s law [8], [49]) we have
N = λW = λ(Wd+ Ws). Substituting this in (10) we obtain
Ws= sλ(Wd+ Ws) +λs 2
This implies Ws= ρ 1 − ρWd+ λs2 2(1 − ρ), (11)
Substituting (9) and (11) in (8) yields (7). ut
In addition to the average waiting time of a classical M/G/1 queue given in (1), the queueing delay also increases due to the collector’s travel.
2.4 Collector Policies
We derive upper bounds on delay by analyzing policies for the collector. The TSPN policy analyzed in the previous section is stable for all loadsρ < 1 and has O(1−ρ1 ) delay scaling. Since the lower
bound in Section 2.3 also scales with the load as 1
1−ρ, the TSPN policy has optimal delay scaling. In the following we consider the First Come First Serve (FCFS) and the Partitioning policies that can have better delay properties than the TSPN policy. In particular, the FCFS policy is delay-optimal at light loads and the Partitioning policy has delay performance that is very close to the lower bound when the travel and reception times are comparable.
2.4.1 First Come First Serve (FCFS) Policy
A straightforward policy is the FCFS policy where the messages are served in the order of their arrival times. A version of the FCFS policy, call FCFS’, where the receiver has to return to the center of the network region (the median of the region for general network regions) after each message reception was shown to be optimal at light loads for the DTRP problem [8]. This is because the center of the network region is the location that minimizes the expected distance to a uniformly distributed arrival. Since in our system we can do at least as good as the DTRP by settingr∗= 0, FCFS’ is optimal also for our system at light loads. Furthermore, the FCFS policy is not stable for all loadsρ < 1, namely,
there exists a valueρ such that the system is unstable under FCFS policy for all ρ > ˆˆ ρ. This is because
under the FCFS policy the average travel component of the service time is fixed, which makes the average arrival rate greater than the average service rate asρ → 1. Therefore, it is better for a policy to serve more messages in the same “neighborhood” in order to reduce the amount of time spent on mobility.
2.4.2 Partitioning Policy
Next we propose a policy based on partitioning the network region into subregions and the collector performing a cyclic service of the subregions. This policy is an adaptation of the Partitioning policy of [8] to the case of a system with wireless transmission. We explicitly derive the delay expression for this policy and show that it scales with the load asO( 1
1−ρ) as in the TSPN policy.
We divide the network region into(√2r∗x√2r∗) squares as shown in Fig. 2. This choice ensures us that every location in the square is within the communication distancer∗of the center of the square. The number of subregions in such a partitioning is given by3n
s = A/(2(r∗)2). The partitioning in Fig. 2 represents the case ofns = 16 subregions. The collector services the subregions in a cyclic order as displayed in Fig. 2 by receiving the messages in each subregion from its center using an FCFS order. The messages within each subregion are served exhaustively, i.e., all the messages in a subregion are received before moving to the next subregion. The collector then receives the messages in the next subregion exhaustively using FCFS order and repeats this process. The distance traveled
3 Note that such a partitioning requires√n
s=pA/(2(r∗)2) to be an integer. This may not hold for a given area A and
a particular choice ofr∗. In that case one can partition the region using the largest reception distance
r∗
< r∗such that this
r∗
√ 2r∗
Fig. 2: The partitioning of the network region into square subregions of side√2r∗. The circle with radiusr∗represents the communication range and the dashed lines represent the collector’s path.
by the collector between each subregion is a constant equal to √2r∗. It is easy to verify that the Partitioning policy behaves as a multiuser M/G/1 system with reservations (see [7, p. 198]) where thenssubregions correspond to users and the travel time between the subregions corresponds to the reservation interval. Using the delay expression for multiuser M/G/1 queue with reservations in [7, p. 200] we obtain, Tpart= λs 2 2(1 − ρ)+ ns− ρ 2v(1 − ρ) √ 2r∗+ s, (12)
whereρ = λs is the system load. Combining this result with (7) and noting that the above expression
is finite for all loadsρ < 1, we have established the following observation.
Observation 1 The time average delay in the system scales asΘ(1−ρ1 ) with the load ρ and the
Parti-tioning policy is stable for allρ < 1.
Despite the travel component of the service time, we can achieveΘ( 1
1−ρ) delay as in classical queuing systems (e.q., G/G/1 queue). This is the fundamental difference between this system and the corresponding system where wireless transmission is not used, as in the latter system the delay scaling with load isΘ( 1
(1−ρ)2) [8]. This difference can be explained intuitively as follows. Denote by
N the average number of departures in a waiting time. It is easy to see from the P-K formula that
in a classical M/G/1 queue,N scales with the load as Θ(1−ρ1 ). We argue that this scaling for N is
preserved in our system but not in [8]. TheWsexpression as a function ofWdin (11) implies that for any given policy with its correspondingWd,N can be lower bounded by λW1−ρd. For the system in [8], the minimum per-message distance the collector moves in the high load regime scales as Ω(√√A
N) [8]. Intuitively, this is due to the observation that the nearest neighbor distance amongN uniformly
distributed points on a square region of areaA scales as √√A
N. Therefore, for this system we have
Wd ≈ NΩ( √ A √ N) ≈ Ω( √
N A) which gives N ≈ Ω((1−ρ)λ2A2). Namely, Wd increases with the load and this results in an extra1/(1 − ρ) scaling in delay in addition to the 1/(1 − ρ) factor of classical G/G/1 queues. However, with the wireless reception capability, the collector does not need to move for messages that are inside a disk of radiusr∗around it. Since a finite (constant) number of such disks cover the network region,Wd can be upper bounded by a constant independent of the system load, for example, for the Partitioning policy an easy upper bound onWdis the length of one cyclic tour around the network. Therefore, in our systemN scales as 1/(1 − ρ) as in classical queueing systems. It is interesting to note that [15] considered the case where messages were transmitted to the collector according to a random access scheme, i.e., transmissions occur with probabilityp in each
time slot. There the delay scaling ofΩ( 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 101 102 103 104 Arrival Rate, λ
Delay Lower Bound
No communication r* = 4.73 r* = 2.66 r* = 1.50
Fig. 3: Delay lower bound vs. network load using different communication ranges for A = 200, β = 2, α = 4, v = 1 and s = 1.
wireless transmission. The reason for this is that in order to have successful transmissions under the random access interference of neighboring nodes, the reception distance should be of the same order as the nearest neighbor distances [15], [23].
2.4.3 Numerical Results-Single Collector
Here we present numerical results corresponding to the analysis in the previous sections. We lower bound the delay expression in (7) usingE[(||U|| − r∗)+] ≥ E[||U||] − r∗, where[||U||] = 0.383√A is the expected distance of a uniform arrival to the center of square region of areaA [8]. Fig. 3 shows
the delay lower bound as a function of the network load for increased values of the communication range r∗ 4. As the communication range increases, the message delay decreases as expected. For heavy loads, the delay in the system is significantly less than the delay in the corresponding system without wireless transmission in [8], demonstrating the difference in the delay scaling between the two systems. For light loads and small communication ranges, the delay performance of the wireless network tends to the delay performance of [8].
Fig. 4 compares the delay in the Partitioning Policy to the delay lower bound for two different cases. When the travel time dominates the reception time, the delay in the Partitioning policy is about
10.6 times the delay lower bound. For a more balanced case, i.e., when the reception time is
compa-rable to the travel time, the delay ratio drops to2.4.
3 Multiple collectors - Interference-Free Networks
In this section we extend our analysis to a wireless network with multiple identical collectors that do not interfere with each other. An arriving message is transmitted when one of the m collectors
comes within the reception distance of the message location and grants access for the message’s transmission. Therefore, at a given time there can be at most m transmissions in the network. We
consider policies that partition the network region intom subregions. Each collector is assigned to one
of the subregions and is allowed to operate only in its own subregion. We call this class of policies the network partitioning policies. In such a case, there is no interference from nodes within the subregion
4 For the delay plot of the no-communication system, the point that is not smooth arises since the plot is the maximum of two delay lower bounds proposed in [8].
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 101 102 103 104 Load, ρ Delay Partitioning Policy−Case−1 Lower Bound−Case−1 Partitioning Policy−Case−2 Lower Bound−Case−2
Fig. 4: Delay in the Partitioning policy vs the delay lower bound forr∗ = 2.2, β = 2 and α = 4. Case-1: Dominant travel time (A = 800, v = 1, s = 2). Case-2: Comparable travel and reception
times (A = 60, v = 10, s = 2).
where the transmission is taking place. The only source of interference can be due to transmissions in other subregions.
The interference-free assumption holds for example if the signaling schemes used in different sub-regions are orthogonal to each other. This can be achieved for example by having different frequency bands for transmissions in different subregions. Furthermore, for networks deployed on a large area, even if orthogonal signaling is not utilized, the interference between subregions can be negligible due to signal attenuation with distance.
Note that the interference-free assumption is consistent for the purposes of a lower bound on delay since the interference from neighboring nodes can only increase the message reception time. Similar to the previous section, we assume a communication ranger∗for each collector and each reception takes times.
3.1 Stability
Here we show thatρ = λs/m < 1 is a necessary and sufficient condition for stability of the system.
3.1.1 Necessary Condition for Stability
A necessary condition for stability of the multi-collector system is given by ρ = λs/m < 1. We
prove this by showing that the system stochastically dominates the corresponding system with zero travel times (i.e., an M/D/m queue, a queue with Poisson arrivals, constant service time andm servers)
similar to Section 2.2.
Theorem 4 A necessary condition for the stability of any policy isρ = λs/m < 1. Furthermore, the
optimal steady state time average delayT∗
mis lower bounded by Tm∗ ≥ λs2 2m2(1 − ρ)− m − 1 m s2 2s+ s, (13)
The proof is similar to the proof of Theorem 1 and is given in Appendix-B. It makes use of the fact that the steady state time average delay in the system is at least as big as the delay in the equivalent system in which travel times are considered to be zero (i.e.,v = ∞).
3.1.2 Sufficient Condition for Stability
Next we establish thatρ < 1 is also sufficient for stability. This can be seen by dividing the network
region intom identical square subregions, and performing a single-collector TSPN policy in each
sub-region5. Since the arrival process is Poisson, each subregion receives an independent Poisson arrival
process of intensityλ/m. Furthermore, each collector performs a TSPN policy independently of the
other collectors. Therefore, using the stability result of the single-collector TSPN policy, the systems in each subregion are stable ifρ < 1. We state this fact in the following theorem:
Theorem 5 The system is stable under the multi-collector TSPN policy for all loadsρ = λs/m < 1.
Note that a similar delay analysis to the single-collector TSPN case shows that the multi-collector TSPN policy hasO( 1
1−ρ) delay scaling with the load ρ.
3.2 Delay Lower Bound
The delay lower bound in (13) neglects the travel component of the delay. Therefore, we provide another lower bound for the optimal delayT∗
mand take their simple average. The following theorem states the second lower bound on delay. It is based on the convexity argument that when the travel component of the waiting time is lower bounded by a constant, the equal area partitioning of the network region minimizes the resulting delay expression over all area partitionings.
Theorem 6 For the class of network partitioning policies, the optimal steady state time average delay
T∗ mis lower bounded by Tm∗ ≥ max 0,2 3 q A mπ− r∗ v(1 − ρ) + s, (14)
whereρ = λs/m is the system load.
Proof Here we use an approach similar to the proof of Theorem 3. We divide the average delayT into
three components:
T = Wd+ Ws+ s. (15)
The lemma below provides a bound forWd, the average message waiting time due to the collectors’s travel, using a result from [25] for the m-median problem.
Lemma 1 Wd≥ max 0,2 3 q A mπ− r∗ v . (16)
Proof LetΩ be any set of points in < with |Ω| = m. Let U be a uniformly distributed location in < independent ofΩ and define Z∗ , min
ν∈Ω k U − ν k. Let the random variable Y be the distance from the center of a disk of areaA/m to a uniformly distributed point within the disk. Then it is shown
in [25] that
E[f (Z∗)] ≥ E[f(Y )] (17)
for any nondecreasing functionf (.). Using this result we obtain E[max(0, Z∗−r∗)] ≥ E[max(0, Y −
r∗)]. Note that W
dcan be lower bounded by the expected distance of a uniform arrival to the closest collector at the time of arrival lessr∗. Because the travel distance is nonnegative, we have
Wd≥ E[max(0, Y − r∗)]/v ≥ max(0, E[Y ] − r∗)/v, where the second bound is due to Jensen’s inequality. SubstitutingE[Y ] = 23
q
A
mπ into the above
expression completes the proof. ut
Intuitively the best a priori placement ofm points in < in order to minimize the distance of a uniformly distributed point in the region to the closest of these points is to cover the region withm disjoint disks
of areaA/m and place the points at the centers of the disks. Such a partitioning of the region is not
possible, however, using this idea we can lower bound the expected distance as in (16).
We now derive a lower bound onWs. LetR1, R2, ..., Rmbe the network partitioning with areas
A1, A2, ..., Amrespectively (Pm
j=1Aj= A). Consider the message receptions in steady state that are received by collectorj eventually. Let λjbe the fraction of the arrival rate served by collectorj. Due to the uniform distribution of the message locations we have
λj
λ = Aj
A .
Let Nj be the average number of message receptions for which the messages that are served by collectorj waits in steady state. Similarly let Wj
s andW j
d be the average waiting times for messages served by collectorj due to the time spent on message receptions and collector j’s travel respectively.
Using (10) and lower bounding the residual time by zero we have
Wsj ≥ sNj. Using Little’s law (Nj= λj(Wj
s + W j
d)) in a similar way as for (11) we have
Wsj≥
λjs
1 − λjsW j
d. (18)
The fraction of messages served by collectorj is Aj/A. Therefore, we can write W sas Ws= m X j=1 Aj AW j s ≥ m X j=1 Aj A λjs 1 − λjsW j d. (19)
For a given regionRj with areaAj,Wj
d is lower bounded by (similar to the derivation of (9)) the distance of a uniform arrival to the median of the region lessr∗.
Wdj≥
E[max(0,||U − ν|| − r∗)]
v ≥
max(0, E[||U − ν||] − r∗)
v , (20)
whereν is the median of Rjand||U − ν|| is the distance of U, a uniformly distributed location inside
Rj, toν. The inequality in (20) is due to Jensen’s inequality for convex functions. A disk shaped region yields the minimum expected distance of a uniform arrival to the median of the region [25]. Using this we further lower boundWdby noting that for a disk shaped region of areaAj,E[||U − ν||] is just the expected distance of a uniform arrival to the center of the disk given by 23
q Aj π. Hence Wdj≥max(0, 2 3 q Aj π − r∗) v = max(0, c1 √ Aj− r∗) v , (21)
wherec1 = 3√2π = 0.376. Letting f (Aj) = λ Aj
As 1−λAj As
, which is a convex and increasing function of
Aj, we rewrite (19) as Ws≥ m X j=1 f (Aj) vA A jmax(0, c 1 √ Aj− r∗). (22)
Next we show that the functionf (Aj)Ajmax(0, c 1
√
Aj− r∗) is a convex function of Ajvia the two lemmas below.
Lemma 2 Letf (.) and g(.) be two convex and increasing functions. The function h(.) = f (.)g(.) is
also convex and increasing.
Proof See Appendix C. ut
Lemma 3 h(x) = x max(0, c1√x − c2) with domain [0, ∞) is a convex and increasing function of
x.
Proof See Appendix D. ut
Lettingg(Aj)= f (A. j)Ajmax(0, c 1
√
Aj− r∗), we have from the lemmas 2 and 3 that the function
g(Aj) is convex. Now rewriting (22) we have
Ws≥ ( m vA) 1 m m X j=1 g(Aj).
Using the convexity of the functiong(Aj) we have
Ws ≥ ( m vA)g Pmj=1Aj m = m vAg( A m) = λs m 1 −λsm max(0, c1 q A m− r∗) v = ρ 1 − ρ max 0, c1 q A m− r∗ v . (23)
The above analysis essentially implies that theWsexpression in (22) is minimized by the equitable
partitioning of the network region. Finally combining (15), (16) and (23) we obtain (14). ut
Finally, taking the simple average of (13) and (14) we arrive at the following theorem.
Theorem 7 For the class of network partitioning policies, the optimal steady state time average delay
T∗ mis lower bounded by Tm∗ ≥ λs2 4m2(1−ρ)+ max 0,2 3 q A mπ−r∗ 2v(1−ρ) − m−1 m s2 4s+ s, (24)
whereρ = λs/m is the system load.
This expression incorporates both the travel time and the message reception time components of the expected delay in the system. Theorem 7 is valid for the class of network partitioning policies. For the system without wireless transmission, it has been shown that partitioning the region intom equal size
disjoint subregions (one for each collector) preserves optimality in the high load limit [9], [51]. We conjecture that this optimality is also preserved in our system.
r∗
√2r∗
Fig. 5: The partitioning of the network region into square subregions of side √2r∗ for the case of multiple collectors in the network. The circle with radiusr∗represents the communication range and the dashed lines represent one of the collector’s path.
3.3 Multiple Collector Policies
It is easy to show that a generalization of the FCFS policy in which we partition the network region intom subregions and assign each collector to perform the single-collector FCFS Policy in its own
subregion has optimal delay at light loads. The reason for this results is similar to the light load optimality of the FCFS Policy for the single collector case. The area partitioning has to be done as follows: We createm Voronoi regions with centers of the regions given by the m-median locations6
of the network region. This policy is not stable asρ → 1 due to the same reason as in Section 2.4. 3.3.1 Generalized Partitioning Policy
Next we propose a policy based on dividing the network region into m equal size subregions. For
simplicity we assume that it is possible to divide the region intom identical square subregions. Each
collector is assigned to one of the subregions and is responsible for receiving messages that arrive into its own subregion using the single collector partitioning policy analyzed in Section 2.4.2. Namely, first the network region is divided into subregions of areaA/m and then each subregion is divided
into√2r∗x√2r∗squares. The number of√2r∗x√2r∗squares in each subregion is given by7n
s= A/m
2(r∗)2. Fig. 5 represents such a partitioning for the case of four collectors in the network withns= 16 squares in each subregion. Since each subregion behaves identically, the average delay of this policy is the average delay of the single collector Partitioning policy applied to a subregion with arrival rate
λ/m, area A/m, and ns=2(rA/m∗)2:
Tpart= λs2 2m(1 − ρ)+ A 2m(r∗)2 − ρ 2v(1 − ρ) √ 2r∗+ s, (25)
whereρ = λs/m is the system load. This result, when combined with (13), establishes that for the
case of multiple collectors in the system the delay scaling with the load isΘ( 1
1−ρ). This is again a 6 The set ofm-median locations for a region is the set of the best m a priori locations in the region that minimizes the expected distance to a uniform arrival.
7 As for the single collector Partitioning policy, we note that if√n
s = p(A/m)/(2(r∗)2) is not an integer, one can
partition the region using the largest reception distancer∗
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 101 102 103 104 Load, ρ
Delay Lower Bound
Multiple Collectors No Communication
With Communication
Fig. 6: Delay lower bound vs. network load for m=2 collectors,r∗ = 4.7, A = 400, β = 2, α = 4,
v = 1 and s = 1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 101 102 103 Load, ρ
Delay Lower Bound
Multiple Collectors
Lower Bound Partitioning Policy
Fig. 7: Delay of the Partitioning policy vs the delay lower bound for m = 4 collectors, r∗ = 2.6,
A = 500, β = 2, α = 4, v = 1 and s = 2.
fundamental improvement compared to theΘ((1−ρ)1 2) delay scaling in the system without wireless transmission and with multiple collectors in [9].
3.3.2 Numerical Results
We compare the delay lower bound in (24) to the delay lower bound in the corresponding system without wireless transmission in [9] for the case of two collectors in Fig. 6. The delay in the two-collector system is significantly below the delay in the system without wireless transmission and this difference is more pronounced for high loads. Fig. 7 displays the delay lower bound in (24) and the delay of the Partitioning policy in (25) as functions of the network loadρ. The delay of the Partitioning
4 Multiple Collectors - Systems with Interference Constraints
In this section we consider systems in which simultaneous transmissions to different collectors in-terfere with each other. Inin-terference between simultaneous transmissions occurs if the collectors do not use orthogonal signaling. At each point in time, the problem is to dynamically determine message pick up locations for the collectors and also to efficiently route the collectors to these pick up locations based on the current collector configuration and the number of messages present in different parts of the network region. The objective is to minimize the expected message waiting time in the system. This is a joint scheduling and euclidian vehicle routing problem which has not been considered previ-ously.
Here we obtain preliminary results for this problem by emphasizing the scheduling aspects of the problem through fairly general interference constraints and simplifying the mobility aspect of the problem by discretizing the collectors’ motion. We characterize the stability region of the system in terms of interference constraints. We show that a frame-based version of the Max-Weight scheduling policy [11], [46], can stabilize the system whenever it is stabilizable and we derive an upper bound on the average delay under this policy.
First we explain the model in more details. Similar to the previous section, considerm collectors
in a square regionR of area A receiving data messages that arrive in time according to a Poisson process of intensityλ. Upon arrival the messages are distributed independently and uniformly in R and an arriving message is transmitted when a collector comes within the reception distance,r∗, of the message location and grants access for the message’s transmission. We assume that time is slotted,
t = 0, 1, 2, ..., where the slot length is equal to one message transmission time s. We consider a
partitioning of the network region into a gridG of √r∗ 2x r∗ √ 2 squares, i.e., K . = A (r∗)2/2square cells of diameterr∗as shown in Fig. 88. The collectors are confined to move on the gridG and simultaneous transmissions to different collectors are subject to interference constraints defined below.
Definition 3 (Cell Interference Model) Given a collector that is at the intersection of (at most) 4
adjacent cells, a transmission to the collector from one of the cells is successfully received if there is no other transmission within the other cells adjacent to the collector.
The Cell Interference Model essentially creates an exclusion region of 4 cells around a collector re-ceiving a message. Similar interference models have been considered in literature. For example, the Protocol Model considered in [24], [43] assumes successful transmission if a disk region around the receiver has no other transmission. Similarly, the Vulnerability Circle Model considered in [16] or the Disk Reception Model considered in [15] require an exclusion region around receivers for successful reception.
The Cell Interference Model essentially createsK cells which can be treated as “users” in a
multi-user queue withm servers. Assume a fixed ordering of these subregions. Due to the splitting property
of the Poisson process, each cell i = 1, 2, ..., K, receives Poisson arrivals with rate λi = λ/K.. Let Ai(t) denote the number of messages that arrive into cell i in time slot t. The expected load entering celli per time slot is given by ρi = λis. Furthermore, let Ni(t) be the number of messages present in cell i at the beginning of time slot t. We have N(t) = [N. 1(t), ..., NK(t)] and N (t) =
N1(t) + ... + NK(t), where N (t) is the total number of messages in the system at the beginning of time slott. We assume that the system is initially empty, therefore N (0) = 0.
Next we characterize the interference constraints of the system in terms of activation vectors. We call a cell active if at least 1 message in the cell is scheduled for transmission, and we assume that each cellk = 1, 2, ..., K is associated with exactly one pick up location on the grid G. For instance, the pick up location for each cell could be the upper left corner of the cell as shown in Fig. 8. Therefore, specifying the set of cells to activate also specifies the locations of the collectors. A feasible activation
8 Similar to the previous sections, ifK = A
(r∗)2/2is not an integer, one can partition the region using the largest reception distancer∗
An Inactive Message
An Active Message Transmission A Server
A Moving Server
r∗
Fig. 8: The network model. Red regions are the exclusion zones for the servers currently in service. 2 Servers are forced to be inactive since the queue in their vicinity are in the exclusion zone of another server.
vector I∈ I is the one under which the transmissions from the set of active cells do not interfere with each other, whereI is the set of all feasible activation vectors. If cell k, k = 1, ..., K, is active under activation vector I, then we have I(k) = 1, otherwise I(k) = 0. The set I consists of K-dimensional vectors of at mostm nonzero entries. Furthermore, because transmissions require an exclusion zone
of up to 4 adjacent cells, the number of nonzero entries of vectors inI is typically less than m. Note that we include the zero vector I= 0 in I for convenience.
An alternative way of modeling the interference constraints is to have a4 by K matrix for each
activation I. In this representation every column of the matrix I corresponds to a single cell and the different rows in a column vector specifies the position of the collector responsible for serving the cell. Therefore, for a given matrix I, at most one element of a column vector can be nonzero. The matrix activation model is more complex, however, it does not require the extra assumption that each cell is associated with a single pick up location. Here we present our results in terms of the simpler vector activation model for ease of exposition.
LetTrdenote the number of time slots required for a collector to move from the lower right corner of the gridG to the upper left corner of G. We call Trthe reconfiguration time of the network. Consider the corresponding system with infinite speed, i.e.,Tr= 0. This system is essentially a parallel queuing system with with multiple servers and interference constraints, which is a special case of the system considered in [38] or [46]. When Tr = 0, the stability region of this system, Λ0, consists of the closure of all load vectors ρ= [ρ1, ρ2, ..., ρK]0 = s[λ1, λ2, ..., λK]0 in the convex hull of the vectors inI [12], [37], i.e.,
Λ0= {ρ|ρ ∈ Conv{I}}. (26)
The celebrated Max-Weight scheduling algorithm was introduced in [46] and was shown to be throughput optimal for the system with zero reconfiguration time. Specifically, the Max-Weight policy activates the set of users in I∗(t) where
I∗(t) = arg max
I∈I
N(t).I. (27)
WhenTr> 0, we lose service opportunities during the reconfiguration times. Therefore, the stability region of our system can be no larger than Λ0:
Lemma 4 An outer bound on the stability region is given by
The analysis in the next section shows that we have Λ = Λ0. The intuition behind this is that, under throughput-optimal policies, as the load approaches the boundary of the stability region, the fraction of time spent to reconfiguration tends to zero.
4.1 Framed-Max-Weight Policy
For systems with nonzero reconfiguration times, the Max-Weight policy is not throughput-optimal [11], [12]. The intuitive reason behind this is that the Max-Weight policy gives decisions to change the schedule too frequently, resulting in throughput-loss during the reconfiguration intervals. A frame based version of the Max-Weight policy where the same schedule is used throughout the frame incurs less throughput loss during the reconfiguration intervals. Indeed, single hop optical networks with in-terference constraints and nonzero reconfiguration times were considered in [11], where a frame-based version of the Max-Weight policy was shown to be throughput-optimal asymptotically in the frame length. Fluid limit of the system was considered in [11] and throughput-optimality was established un-der rate stability9. Furthermore, anN xN switch with matching constraints and reconfiguration delays
was considered in [12], where a policy based on servicing batches of arrivals over frames accord-ing to the Max-Weight activation vector was shown to be throughput optimal asymptotically in the frame length. In fact, we show that the frame-based Max-Weight (FMW) policy proposed in [11] is throughput-optimal asymptotically in the frame length also for the system considered in this section. Different from the analysis in [11] and [12], we prove this result using classical quadratic Lyapunov drift techniques. The reason that the FMW policy stabilizes the system is that as the system load approaches the boundary of the stability region, the policy employs good schedules, i.e., maximum-weight schedules, over longer frame lengths, effectively decreasing the fraction of throughput lost to reconfiguration to zero.
Under the FMW policy the time is divided into intervals of lengthT slots. The FMW policy picks
the activation vector corresponding to the Max-Weight configuration at the beginning of each frame. Then it idles the system forTrslots to ensure that all the servers travel to their assigned locations. Then the policy applies this activation vector for T − Tr slots. The choice of the frame lengthT depends on the load ρ. Specifically, the policy requiresT > Tr/ where is determined by solving the Linear Program below [11].
(ρ), max 1 −X I∈I αI ! subject to ρi= λis ≤ X I∈I αII(i), i ∈ {1, ..., K} X I∈I αI≤ 1 αI≥ 0, ∀I ∈ I. (28) Note that is a measure of distance of the load vector to the boundary of the stability region [11]. The
FMW policy is described in Algorithm 2 in detail.
Theorem 8 For any ρ= [ρ1, ..., ρK]0strictly inside Λ0, the FMW policy stabilizes the system as long
asT > Tr .
The proof is in Appendix E. A similar result is proved in [12] for the fluid limit of this system establish-ing the rate stability. The proof in Appendix E is based on a quadratic Lyapunov drift argument over
9 A queue of lengthN
i(t) at time t is rate stable if limt→∞Ni(t)/t = 0. This is a weaker notion of stability as compared
Algorithm 2 Framed-Max-Weight Policy
1: Find the Max-Weight configuration for the servers based on the queue lengths at the beginning of the frame: Assuming the system is at thekth frame, find
I∗(t) = arg max
I∈I
N(kT ).I. (29)
2: Let the users be idle forTrslots and reconfigure the collectors to their new locations. 3: Apply the activation vector I∗forT − Trslots whereT > Tr.
a sufficiently long frame of durationT . Let tk be the first slot of thekth frame. The proof establishes that theT -step expected drift of the queue lengths satisfies
EL(N(tk+ T )) − L(N(tk)) N(tk) ≤ KBT2− 2T K( − Tr T ) X i Ni(tk), (30) whereB = 1 + λsK + λK2s22 is a constant. This means that the queue sizes tend to decrease if they are outside a bounded set for a sufficiently large frame length which implies bounded expected queue sizes [37].
The following corollary follows from Theorem 8 and Lemma 4.
Corollary 1 We have
Λ= Λ0.
Note that the overall arrival rate λ is given by λ = λ1 + ... + λK where λk = λ/K for k =
1, ..., K. Therefore, the necessary and sufficient stability condition on ρ = λs is ρ < ˆρ where ˆρ is the
intersection of theK-dimensional region Λ with the line ρ1= ... = ρK = ρ/K.
4.2 Delay Analysis
In this section we derive an upper bound on the expected number of messages in the system. Note that through Little’s law, the expected delay in the system is proportional to the expected number of messages in the system. Taking the expectation of (30) over the distribution of N(kT ) we have
E [L(N(tk+ T ))] − E [L(N(tk))] ≤ KBT2− 2T K( − Tr T ) X i E [Ni(tk)] .
We write a similar drift expression fork = 0, 1, ..., M − 1, and sum over k to obtain a telescoping series that gives
E [L(N(tM))] − E [L(N(0))] ≤ MKBT2− 2T K( − Tr T ) M−1 X k=0 X i E [Ni(tk)] ,
wheretk+1 = tk+ T , k = 0, 1, .... Dividing by M and using L(N(tM)) ≥ 0 and L(N(0)) = 0 we have 1 M 2T K( − Tr T ) M−1 X k=0 X i E [Ni(tk)] ! ≤ KBT2. Taking thelim sup as M → ∞ we have
lim sup M→∞ 1 M M−1 X k=0 X i E [Ni(tk)] ≤ K2T B 2( − Tr/T ).
Noting thatP
iNi(t) = N (t) that is the total number of messages in the system at time slot t we have
lim sup M→∞ 1 M M−1 X k=0 E [N (tk)] ≤ K 2T B 2( − Tr/T ) . (31)
Now for any givent ∈ {tk, tk+1− 1} we have
N (t) ≤ N(tk) + T −1 X τ =0 X i Ai(tk+ τ ).
Summing over the time slots within thekth frame and taking conditional expectation we have for any t ∈ {tk, tk+1− 1} T −1 X τ =0 E [N (tk+ τ )|N(tk)] ≤ T N(tk) + T2 X i λis,
where we used the fact that arrival processes are i.i.d. and independent of the queue lengths. Using
λ = P
iλi, taking expectation with respect to N(tk), writing similar expressions for M frames and summing them we obtain
1 M T MT −1 X t=0 E [N (t)]≤M1 M−1 X k=0 E [N (tk)] + T ρ. Finally, taking thelim sup as M → ∞ and using (31) we have
lim sup t→∞ 1 t t−1 X τ =0 E [N (t)]≤ K 2T B 2( − Tr/T ) + T ρ,
Using this expression it can be shown that [37]
lim sup
t→∞ E [N (t)]≤
K2T B
2( − Tr/T )+ T ρ.
For large frame lengths such thatT >> Tr, the delay scaling is approximately proportional to1/, where is a measure of the distance of arrival rate to the boundary of the stability region. Note that
when the reconfiguration interval is large, large frame lengthsT >> Trmay be required for achieving throughput-optimality, which, on the other hand, increases the expected delay as the delay under the FMW policy is proportional to the frame length.
5 CONCLUSION
In this paper we considered the use of dynamic vehicle routing in order to improve the throughput and delay performance of wireless networks where messages arriving randomly in time and space are gathered by mobile collectors via wireless communications. For the case of a single collector, we characterized the stability region of this system to be all system loads ρ < 1. We developed
fundamental lower bounds on time average expected delay and derived upper bounds on delay by analyzing TSPN and Partitioning policies. For the case of multiple collectors whose communications do not interfere with each other, we extended the stability and delay scaling results of the single collector case. Our results show that combining controlled mobility and wireless transmission results inΘ( 1
1−ρ) delay scaling with load ρ. This is the fundamental difference between our system and the system without wireless transmission (DTRP) analyzed in [8] and [9] where the delay scaling with the load isΘ( 1
interfere with each other we formulate a scheduling problem and characterize the stability region of the system in terms of interference constraints. We show that a frame-based version of the Max-Weight policy is throughput-optimal asymptotically in the frame length and derive an upper bound on average delay under this policy.
This work is a first attempt towards utilizing a combination of controlled mobility and wireless transmission for data collection in stochastic and dynamic wireless networks. Therefore, there are many related open problems. In this paper we have utilized a simple wireless communication model based on a communication range. In the future we intend to study more advanced wireless communi-cation models such as modeling the transmission rate as a function of the transmission distance. For the case of multiple-collectors whose transmissions are subject to interference constraints, we intend to study interference models that do not restrict the collectors’ motion to a grid. Note that such a joint server routing and scheduling problem is significantly more involved. For instance, the stability region of such a system depends on the interference constraints, and it is unknown since there are uncountably many possible activation vectors.
Appendix A - Proof of Theorem 1
We first show that the unfinished work and the delay experienced by a message in the system stochas-tically dominates that in the equivalent system with zero travel times for the collector.
Lemma 5 The steady state time average delay in the system is at least as big as the delay in the
equivalent system in which travel times are considered to be zero (i.e.,v = ∞).
Proof Consider the summation of per-message reception and travel times,s and di, as the total service requirement of a message in each system. Sincediis zero for alli in the infinite speed system and since the reception times are constant equal tos for both systems, the total service requirement of each
message in our system is deterministically greater than that of the same message in the infinite speed system. LetDiandD
0
i,i = 1, 2..., be the departure instant of the ith message in the original and the infinite speed system respectively. Similarly letAi,i = 1, 2... be the arrival time of the ith message in both systems. We will use induction to prove thatDi ≥ D
0
ifor alli. We trivially have D1 ≥ D 0 1. Furthermore,
An+1≤ Dn+1− s, (32)
hence then + 1th message is available before the timeD
n+1− s. Using the induction hypothesis,
Dn≥ D 0
n, we have
Dn0 ≤ Dn≤ Dn+1− s,
where the second inequality is because we need at leasts amount of time between the nthandn + 1th transmissions. Hence the collector is available in the infinite speed system before the timeDn+1− s. Combining this with (32) proves the induction.
Now letD(t) and D0(t) be the total number of departures by time t in our system and the infinite
speed system respectively. Similarly letN (t) and N0(t) be the total number of messages in the two
systems at timet. Finally let A(t) be the total number of arrivals by time t in both systems. We have A(t) = N (t) + D(t) = N0
(t) + D0
(t). Using the result of the above induction we have D(t) ≤ D0
(t)
and therefore
N (t) ≥ N0(t).
Since this is true at all times, we have that the time average number of customers in the system is greater than that in the infinite speed system. Finally using Little’s law proves the lemma. ut Since the infinite speed system is an M/G/1 queue (an M/G/1 queue is a queue with Poisson arrivals, general i.i.d. service times and 1 server and an M/D/1 queue has constant service times), the average waiting time in this system is given by the Pollaczek-Khinchin (P-K) formula for M/G/1 queues [7, p.
